# An Efficient Global Algorithm for Single-Group Multicast Beamforming

## TL;DR

This paper introduces the first tailored global branch-and-bound algorithm for single-group multicast beamforming, guaranteeing optimal solutions and outperforming existing methods in efficiency and scalability.

## Contribution

It presents a novel, guaranteed global optimization algorithm specifically designed for the NP-hard multicast beamforming problem, setting a new benchmark.

## Key findings

- The algorithm guarantees finding the global solution.
- It outperforms the Baron solver in computational efficiency.
- Performance of existing algorithms declines with larger problem sizes.

## Abstract

Consider the single-group multicast beamforming problem, where multiple users receive the same data stream simultaneously from a single transmitter. The problem is NP-hard and all existing algorithms for the problem either find suboptimal approximate or local stationary solutions. In this paper, we propose an efficient branch-and-bound algorithm for the problem that is guaranteed to find its global solution. To the best of our knowledge, our proposed algorithm is the first tailored global algorithm for the single-group multicast beamforming problem. Simulation results show that our proposed algorithm is computationally efficient (albeit its theoretical worst-case iteration complexity is exponential with respect to the number of receivers) and it significantly outperforms a state-of-the-art general-purpose global optimization solver called Baron. Our proposed algorithm provides an important benchmark for performance evaluation of existing algorithms for the same problem. By using it as the benchmark, we show that two state-of-the-art algorithms, semidefinite relaxation algorithm and successive linear approximation algorithm, work well when the problem dimension (i.e., the number of antennas at the transmitter and the number of receivers) is small but their performance deteriorates quickly as the problem dimension increases.